R is a positive number and and are complex numbers. Show that the points which represent respectively the numbers , , form the vertices of a right angled triangle.

Let .

The displacement from z_1 to z_3 is given by .

The displacement from z_2 to z_3 is given by .

Therefore . So to get from to you have to multiply it by R (which changes its length but not its direction) and then multiply it by i (which has the effect of rotating it through a right angle. This shows that the line from z_1 to z_3 is perpendicular to the line from z_2 to z_3.